Problem 10 - Elementary and Intermediate Algebra

Question

Express in terms of \(i\) $$ \sqrt{-25} $$

Step-by-Step Solution

Answer

5i

Step 1: Identify the given expression

The given expression to simplify is \ \( \sqrt{-25} \ \). Notice that the expression is a square root of a negative number.

Step 2: Recall the definition of imaginary unit

The imaginary unit is defined as \( i = \sqrt{-1} \).

Step 3: Rewrite the expression using the imaginary unit

Rewrite \( \sqrt{-25} \) as \( \sqrt{25} \cdot \sqrt{-1} \). This allows the expression to be split into two separate square roots.

Step 4: Evaluate the square roots

Calculate \( \sqrt{25} = 5 \) and \( \sqrt{-1} = i \).

Step 5: Combine the results

Combine the results from the previous step to get \( \sqrt{-25} = 5i \).

Key Concepts

Understanding Square Roots

The square root is a number that, when multiplied by itself, gives the original number. For instance, the square root of 9 is 3, because 3 multiplied by 3 equals 9. The square root is denoted by the radical symbol \(\sqrt{ }\), and the number under the radical is called the radicand.
However, when dealing with square roots of negative numbers, things become more interesting. This leads us to the concept of imaginary numbers, which we'll discuss shortly.

Dealing with Negative Numbers

Negative numbers are numbers less than zero, commonly found on the left side of the number line. They are crucial in various mathematical operations and real-world scenarios, such as temperature below freezing point or depths below sea level.
When we talk about the square roots of negative numbers, traditional arithmetic doesn't give a real number solution. For example, there is no real number that, when multiplied by itself, equals -1. This is because positive numbers always result in a positive product, and negatives do the same but result in a positive product later through the values. This contradiction led to the invention of imaginary numbers.

Introducing Complex Numbers

Complex numbers are an extension of the traditional number system. They consist of a real part and an imaginary part. An imaginary number is defined as the square root of a negative number. The most basic imaginary number is represented by the unit \(i\) where \(i = \sqrt{-1}\).
In the case of\( \sqrt{-25}\), we can break it down using \(i\): \( \sqrt{-25} = \sqrt{25}\cdot \sqrt{-1} = 5 \cdot i = 5i\). Hence, the combination of real numbers and imaginary numbers forms complex numbers, written as \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part.